Stability of Exponential Euler Method for Linear Stochastic Delay Differential Equations

Abstract

The main purpose of this paper is to investigate the exponential stability in mean square of the exponential Euler method to linear stochastic delay differential equations (LSDDEs). The classical stability theorem to LSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to LSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to LSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to LSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.